Write down the Gibbs sum. Be sure to get all of the indices correct.

Give the expression for the absolute probability that a system will be found in the state with N₁ particles and energy .

Give an expression for the average value of a property A for a system in diffusive and thermal contact with a "reservoir". A "reservoir" is a huge system next to our smaller system with large energy and number of particles.

Give an expression for the average number of particles in a system that is in thermal and diffusive contact with a reservoir.

Suppose that we have a system that can be unoccupied or can have one particle in a state with energy . Write the Gibbs sum for this system.